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A-LevelMathematicsAlgebra9709/22 · Oct/Nov 2021 · Q18 Marks

The polynomial p(x) is defined by p(x) = ax³ + bx – 10, where a and b are constants. It is given that (x + 2) is a factor of p(x) and that the remainder is –55 when p(x) is divided by (x + 3).

  1. (a)Find the values of a and b.[5]
  2. (b)Hence factorise p(x) completely.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

  1. [*M1] Substitute x = –2 and equate to zero
  2. [*M1] Substitute x = –3 and equate to –55
  3. [A1] Obtain –8a – 2b – 10 = 0 and –27a – 3b – 10 = –55 OE
  4. [DM1] Solve a pair of relevant simultaneous linear equations to find a or b
  5. [A1] Obtain a = 4 and b = –21
  6. [M1] Divide by x + 2 at least as far as the x term Or using identity or inspection.
  7. [A1] Obtain 4x² – 8x – 5
  8. [A1] Conclude (x + 2)(2x + 1)(2x – 5) Condone inclusion of = 0.

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About this question

Topic
Algebra, A-Level Mathematics (syllabus 9709). Structured question, 8 marks.
Source
A-Level Mathematics, Oct/Nov 2021 Paper 2 Variant 2, question 1.

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